(Hnge- und Spannbandbrcken) 12.05.2020 ETH Zrich | Chair of - - PowerPoint PPT Presentation
Suspension Bridges (Hnge- und Spannbandbrcken) 12.05.2020 ETH Zrich | Chair of Concrete Structures and Bridge Design | Bridge Design 1 Common aspects Suspension bridges Overview Suspension bridges (types of suspension bridges)
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Suspension bridges
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Common aspects Cable-stayed bridges
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Suspension bridges Cable system Stiffening girder Towers stress-ribbons (suspended bridges) Anchor blocks Overview (types of suspension bridges)
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cables can be categorised as follows: → Suspension bridges: Strongly sagging main cables spanning between towers. Cables loaded laterally by vertical hangers connecting the suspended deck girder to the main cables. → Suspended bridges / stress-ribbons: Slightly sagging main cables, spanning between abutments without towers. Cables loaded laterally by the deck girder. The deck follows the cable profile in elevation.
stress-ribbons if the deck consists of a prestressed concrete slab. However, the term “stress-ribbon” is also used for other types of suspended bridges.
suspended bridges / stress-ribbons differ by an order
Suspension bridge (Hängebrücke)
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Suspended bridge / Stress-ribbon (Spannband)
main cables tower / pylon hangers / suspender cables deck / stiffening girder anchor block scale differs by an
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presumably among the first bridges mankind used.
the main cables: → very flexible structures under non-funicular loads (see section static analysis of cables) → range of application very limited: Trails, pedestrian bridges with alternative routes (wheelchairs), etc.
and built with moderate technical know-how unless spans are very long (such as in the Randa bridge designed by Theo Lauber, with a span of 494 m, equipped with special damping devices).
partly for access in mountain areas, partly as mere tourist attractions.
established by Helvetas more than 50 years ago, see next slide.
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Nepal in 1956. Since then, more than 7’000 trail bridges have been built, with suspended bridges up to spans of 156 m, and suspension bridges up to 355 m (see notes for details).
vocational training and trail bridge programs to practical activities reducing communities' vulnerability to disasters.
see www.helvetas.org.
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reasons by aerodynamic stability (overturning of “deck”, as e.g.
Nepal and in Switzerland. Some of them are spectacular, such as the Panoramabrücke Sigriswil with a span of 344 m, 85 m above ground (Martin Dietrich, Theiler Ingenieure).
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bridges carrying road and/or rail traffic (upper photo).
discussed (lower photo).
treated in more detail in the lecture.
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the specific site. Preferences of clients and designers are also important due to the high visual impact of long-span bridges.
which differ mainly in the following aspects:
(construction process!) and common sag/span ratios range from 1/8…1/11, with the following advantages of large/small sag:
→ lower cable forces = savings in cables and anchorages
→ stiffer cables = reduced deck girder bending moments, better aerodynamic behaviour → shorter towers and more elegant appearance
see Gimsing 2012), but deflections under traffic loads are excessive at such large sags (see static analysis of cables).
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Earth-anchored without side spans side spans suspended side spans
side spans suspended girder simply supported girder continuous Self-anchored side spans suspended side spans
side spans suspended
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based on a parabolic cable geometry and the dead load sag :
estimated, requiring iteration. Knowing the cable strength fsd and its specific weight gm (total cable weight per length / steel cross- sectional area), the equation can be solved for the required steel area Am: The required hanger cross-section can be estimated by attributing to each hanger the uniformly distributed load (including traffic loads) corresponding to its part of the deck surface, assuming that concentrated loads are distributed over a length of 30d:
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Estimation of suspension cable force (parabolic sag)
l q g f
H sh sh sh sh d
h
F Q Estimation of hanger forces [Gimsing 2012] 4 f H l Q
q g
( ) ( ) ( )
2 2 2 2 2 2
2 8 4 8 16 16 2 8
m m m
g g q l g g q l Q Ql H f f f l f l f T H g g q l Q l f + + + + + = + = + + = + + +
( )
2 2 2 2
2 16 , 8 16
m sd m m m m sd m
g q l Q l f T A f g A A f f l l f + + + = = g → − g + 30
h h h h sd
T Q T g q s A d f = + + →
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cable-stayed bridges where this is the common solution:
cable force to the stiffening girder at their ends
as the horizontal component of the cable force
advantages and drawbacks:
after stiffening girder is continuous, similar as in tied arches)
was popular e.g. in Germany during the first decades of the 20th century, only few major self-anchored suspension bridges have been built, all of them with moderate spans.
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Earth-anchored without side spans side spans suspended side spans
side spans suspended girder simply supported girder continuous Self-anchored side spans suspended side spans
side spans suspended
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(1990) is an example of an efficient (in the final state) and aesthetically appealing self-anchored suspension bridge.
erecting major self-anchored suspension bridges – essentially, two bridges need to be built.
suspension bridge is the eastern section of the San Francisco-Oakland Bay bridge, which replaced the existing truss bridge in 2017, with a main span of 385 m.
design contest seeking a signature bridge, marks “an extreme in complications during design and construction” according to Gimsing (2012). The final cost of $6.5 billion – 25 times higher than the initial cost estimate – substantiates this criticism.
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stiffness under vertical traffic loads in the main span lm, see section static analysis of cables, since the end span cables control the displacements of the tower top → the stiffness decreases with the length of the side span and the sag in these spans (girder weight).
… highest stiffness … suitable if approaches on land are high enough
… high stiffness … common solution
… low stiffness … aesthetically pleasing
… very low stiffness … stiffening girder partly supported on “columns”
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Second Bosporus bridge: No side spans Brooklyn bridge: Extreme side spans, ls / ls = 0.59
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traffic loads (half main span loaded) is often critical: Under such loads, the cables shift to the side with higher load (see section static analysis of cables, slide on effect of guy cables).
same side), but also the torsional stiffness (cables on either side
central clamp is often provided to ensure a stiffer behaviour using the same effect as that of a guy cable.
torsional stiffnesses are increased by the clamp. While this is favourable for the vertical stiffness, it induces thermal restraint in the cable system (differential temperature of deck and cables), which may require special measures (such as devices permitting slow longitudinal movements, rather than fixed supports).
increases the torsional stiffness. This may be favourable for the behaviour under wind loads (higher ratio of torsional / vertical frequency, see wind-induced oscillations section).
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Deflection under traffic load in right half span (movable cable) z
q g (- - - dead load geometry not drawn to scale) ( ) w q ( ) u q no connection → either cable free to shift → no restraint with clamp, deck long. movable → cables may shift in same direction → stiffer in torsion only with clamp, deck fixed → cables individually restrained → stiffer in bending and torsion
Different midspan connection types between cables and deck
clamp clamp longitudinally fixed
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in elevation, simple solutions are feasible for the central clamps (top left, connection of cable and top chord of stiffening girder truss in the 25 de Abril Bridge, Lisbon).
stiff (top right, Lillebaelt Bridge) or flexible (bottom, Bisan Seto Bridge).
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Central clamps at midspan: Examples
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the cable system if the stiffening girder is longitudinally movable at both ends. In the common case with vertical hangers, this involves a longitudinal displacement of the stiffening girder (longitudinal force resisted by inclination of hangers, short hangers carry most of longitudinal load) → feasible if longitudinal forces are moderate (road bridges)
following options may be chosen: → provide a central clamp (no longitudinal fixity of the stiffening girder required) → provide longitudinal fixity of the stiffening girder at one of the towers → provide hydraulic devices – actuators with a small bypass valve, permitting slow longitudinal displacements without restraint but blocking fast movements – at towers
to limit thermal restraint in the cable system (differential temperature of deck and cables), see previous slides.
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Longitudinal force transfer by short inclined hangers Example of hydraulic buffers at anchor blocks and central clamp (Storebaelt bridge [Gimsing 2012])
x
F 2
x
F H − 2
x
F H +
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hangers, such as in the Severn and Humber bridges, as well as the first Bosporus bridge – all designed by Freeman, Fox &
enhance the aerodynamic stability of slender decks.
deck self weight – do not decompress, a truss-like behaviour is achieved, similar to that in a Nielsen arch (see arch bridges chapter). In the 1960s, even network-suspension bridges had been proposed. If one main cable is used, and the hangers are connected to the outside of the deck, a triangular “truss- box girder” with very high torsional stiffness is achieved.
and the stress range in the hangers increases, which may cause fatigue problems.
achieved by other means, inclined hangers have essentially been abandoned in suspension bridges after the First Bosporus bridge, even in streamlined decks.
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Suspension bridges with Inclined hangers and hanger network
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substantial part of the transverse horizontal loads (wind, seismic loads), even if the cable planes are vertical.
“pendulum effect”: A horizontal displacement v of a cable by a vertical load Fz requires a transverse deviation force
(see historical perspective section) is based on this effect, generalised to distributed loads.
lateral deflection, the contribution of the cable system to the horizontal load transfer can be modelled (in analyses not accounting for large deformations) by horizontal springs with a stiffness
stiffness of the cable system must be reduced accordingly.
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y x
z
F H H z z
z
F v v y
y z
v F F h =
z
F v y h h 4
y z
v F F l =
y z
v F F h =
y y y
g v k q k v g h h = → = =
Contribution of cable system to transfer of transverse loads l x
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several meters are required to resist the full wind load by the pendulum effect of the cable system alone.
loads by bending (horizontal shear, bending moments around vertical axis).
foundation (bottom figure, illustrated for cable-stayed bridge), i.e., the contribution of the cable system is dominant at large
moments in the deck are not significantly affected by the span (almost equal for 600 or 1’200 m span).
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Moments due to lateral wind load on a suspension bridge with a 17 m wide, streamlined deck [Gimsing 2012]
y
g k h =
h
bending moment without contribution
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exist: The deviation forces of the cables (tension) are equilibrated by equal deviation forces of opposite sign in the stiffening girder (compression), see figure.
stayed bridges), the cables arranged in vertical planes do not contribute to the transverse load transfer.
transverse load transfer by truss action, may be used (examples see below, cables must not decompress).
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y x
z
F H H z z
z
F v v y
z
F v y h h Contribution of cable system to transfer of transverse loads l 4
y z
v F F l = 4
y z
v F F l = − z H H x
y
F z y
y
F
z
F
z
F y x x
deviation forces in stiffening girder and cables cancel out
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hangers), the stiffening girder is not carrying substantial axial loads – a significant difference to cable stayed bridges.
→ distribute concentrated loads → carry the load locally between cable anchor points → assist the cable system in carrying the load globally
the global load carrying behaviour is limited, since the cable system is stable by itself → stiffening girder mainly used to limit deformations → support conditions decisive
girders are common in suspension bridges.
no horizontal loads (left photo). Horizontal reactions are resisted by separate wind bearings (right photo).
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Vertical support of suspension bridges
conventional three-span suspension bridge continuous girder supported at towers continuous girder suspended at towers vertically, … but supported in transverse direction elevation elevation elevation plan
Vertical support at tower (“end links” [Gimsing 2012]) Lateral support at tower (“wind bearings” [Gimsing 2012])
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moments particularly at the tower supports.
in the Storebaelt bridge (photo).
lateral deflections. This can be achieved using vertical sliding bearings (below, left figure).
Without vertical support, this is challenging (below, right figure).
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Lateral support at tower using sliding bearings [Gimsing 2012] Lateral and torsional support at tower using sliding bearings [Gimsing 2012]
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provided with orthotropic steel decks. The higher cost compared to concrete decks is compensated by savings in the cable system and erection.
governed by the use of the bridge: → type of traffic and required number of traffic lanes → single or double deck → stiffness requirements (train bridges)
stability – are decisive: → shape: streamlined box or bluff truss girder → torsional stiffness: open or closed cross-section
common in suspension bridges, closed cross-sections are used today to ensure a high torsional stiffness except for short spans (where cable-stayed bridges are more economical).
examples on the right (and many other slides).
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Severn bridge (1966, span 978 m) Lillebælt bridge (1970, span 600 m) Bisan-Seto suspension bridges (1988, spans 990/1100 m
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whereas pylon is used for cable-stayed bridges. In practice, either term may be used for both bridge types.
provided with a high lateral stiffness. Other than in cable-stayed bridges, where pylons are often slender, second order effects (geometrical nonlinearities) are thus of minor importance.
→ loads originating from the deviation of the main cables at the top of the tower (primarily vertical load, governing design) → support reactions of the stiffening girder → wind loads acting on the tower → tower self-weight
towers may be preferred due to other criteria (erection procedure, designers preferences, …)
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Portal-type pylon supporting an earth anchored suspension bridge [Gimsing 2012]
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whereas pylon is used for cable-stayed bridges. In practice, either term may be used for both bridge types.
provided with a high lateral stiffness. Other than in cable-stayed bridges, where pylons are often slender, second order effects (geometrical nonlinearities) are thus of minor importance.
→ loads originating from the deviation of the main cables at the top of the tower (primarily vertical load, governing design) → support reactions of the stiffening girder → wind loads acting on the tower → tower self-weight
towers may be preferred due to other criteria (erection procedure, designers preferences, …)
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Portal-type tower with vertical legs connected by cross-beams [Gimsing 2012] Diagonally braced tower with vertical legs [Gimsing 2012]
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whereas pylon is used for cable-stayed bridges. In practice, either term may be used for both bridge types.
provided with a high lateral stiffness. Other than in cable-stayed bridges, where pylons are often slender, second order effects (geometrical nonlinearities) are thus of minor importance.
→ loads originating from the deviation of the main cables at the top of the tower (primarily vertical load, governing design) → support reactions of the stiffening girder → wind loads acting on the tower → tower self-weight
towers may be preferred due to other criteria (erection procedure, designers preferences, …)
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Portal-type tower: Humber Bridge Diagonally braced tower: Akashi-Kaikyo bridge
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the towers in any case: → centre cable planes in vertical leg axes (see previous slide) → ensure passage of full traffic lanes (upper figure) → pedestrian lanes may pass outside tower legs
to provide passage to the full width of the cross-section. The following tower geometries enable this: → slightly inclined legs, cross-beam at top to deviate cable forces in leg direction (bottom left figure) → vertical legs, saddles positioned eccentrically (with respect to leg axes) on stiff cross-beam ensuring load transfer
the girder can be omitted if the girder is continuously supported by hangers (e.g. Storebaelt bridge, see previous slides).
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Tower geometries for continuous stiffening girder [Gimsing 2012] Simply supported stiffening girder detail at towers [Gimsing 2012]
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suspension bridges are stabilised by the side-span cables. Hence, the buckling length of the towers corresponds to roughly 70% of their height.
bridges are thus relatively slender in the longitudinal direction.
bending moments in the tower due to side span cable
stages must be guaranteed.
longitudinally, having a width of 1/20…1/25 of the tower height.
cladded with marble, since stone was considered appropriate). Later on, steel towers became standard, often using highly complex, structurally inefficient cross-sections (figure).
cell hollow cross-sections, either in steel or concrete (figure).
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Tower of Verrazzano Narrows Bridge (1964) [Gimsing 2012] Tower of Storebælt bridge (1998) [Gimsing 2012]
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Saddle types [Gimsing 2012] (right: movable during construction, fixed in final stage)
top by means of saddles. This is also possible in cable-stayed bridges, where the cables are, however, more often anchored at the pylon top (see cable-stayed bridges section).
cables, and are commonly fixed horizontally to the tower (top left).
and saddle (hence, cable) are sometimes enabled (top right) in
pressure on each strand of the suspension cable, which is limited to avoid reductions of the axial cable strength (particularly fatigue).
→ 0.7…1.8 kN/mm for parallel wire strands and → 1.0…2.0 kN/mm for locked-coil strands (the higher value applies if soft metal sheaths 2 mm are inserted between strand and saddle, or a thick galvanising 2 mm is provided)
Tower saddle of Third Bosporus Bridge
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Frictional forces on cable passing over simple saddle [Gimsing 2012]
and the horizontal component of suspension cable forces on either side of the saddle are thus (approximately) equal, Hl = Hr.
side span, the cable force varies, Tl < Tr for jl > jr. The differential cable force Tr −Tl is transferred by friction, and the maximum force Tr for a given value of Tl is thus: (for derivation see lecture Stahlbeton II, Reibungsverluste).
cable inclinations differ strongly, such as in bridges with short side spans, a cover with pre-tensioned bolts may be pressed against the cable. If m bolts with a preload Pb are used, one gets:
strands in the side span, see lower figures. Note that the horizontal component of the cable force is still approximately constant.
Frictional forces on cable passing over saddle with cover [Gimsing 2012] H H H H
l
T
l
T
r
T
r
T
( )
( )
,max
1
l r
r l l l r
T T e T
m j +j
= + m j + j
( )
( )
,max
2 1 2
l r
r l b l l r b
T T e mP T mP
m j +j
= + m + m j + j + m
Saddle with additional strands for side span cable, anchored
H H
l
T
r
T
main span side span→
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Splay saddle: Elevation and example (Hardanger bridge) [Gimsing 2012]
transferred to the soil through anchor blocks, commonly made from concrete.
split into its strands by means of a splay saddle (Spreizsattel).
arranged in the order they are added during erection. In many cases, they are also flared horizontally, requiring a double curvature of the splay saddle grooves.
the cable due to thermal expansion and contraction (of the splayed strands). In recent bridges (Storebælt, Hardanger, 3rd Bosporus bridge,…), splay saddles were designed as large pendulums for this reason.
thermal movement length strand anchorage deviation force cable force splay saddle splay saddle location
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Splay chamber of 3rd Bosporus Bridge [Klein+Delémont, 2016]
chamber, and are anchored at the bottom of this chamber by means of strand shoes or sockets.
rods, eye-bars or tendons embedded in the concrete.
anchored by looping them around eyebars (example: Tacoma Narrows Bridge, 1950).
splay saddle splay chamber
1
G
2
G R T
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Gravity-type anchor block of Storebælt bridge [Gimsing 2012], with schematic forces
transferring the cable forces – together with the anchor block self weight – to the ground (upper figure)
ballast may be used.
embedded in the rock (lower figure).
structures with a high visual impact and need to be designed carefully.
G
1 1
heavy ballast (iron ore, olivine) ballast (sand) G G
Rock anchorage, Firth of Forth Suspension bridge [Gimsing 2012]
gravel wedges
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g
f l ( ) [m] w q 5 0.1 0.2 traffic load over middle 40% of span traffic load over 100 % of span 0.3 0.4 typical sag (l/11…l/9)
Deflection under traffic load acting over varying length b
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in elevation, and the traffic loads act directly on the suspension cables. Essentially, in elevation, such bridges behave as cables.
(selected illustrations repeated on right) cables are → stiff under funicular loads (loads for which the cable’s initial geometry is funicular, commonly dead load) → very flexible structures under non-funicular loads
limiting their field of application to trail and pedestrian bridges with alternative routes (wheelchairs).
load can be enhanced by → increasing cable tension (see section static analysis
… adding weight or … reducing sag → adding bending stiffness (deck, stiffening girder)
Deflection under traffic load in right half span z x
1000 m l = 2 l 80 kN/m q = 220 m 11 / 44 kN g = (- - - dead load geometry not drawn to scale) ( ) w q w(q) = 7.03 m w(q) = 4.27 m w(q) = 2.38 m
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reasonably stiff deck (usually prestressed concrete) which: → increases the cable tension (weight of concrete) and → adds bending stiffness to the system.
stress-ribbons are stiff enough to be used as footbridges, satisfying the respective serviceability criteria.
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Geometry of a stress-ribbon under uniform load (parabolic geometry)
environmental impact, both visually as well as materially: → slender, minimalist appearance → low material consumption → erection without falsework or shoring affecting the natural environment
the requirement of very high horizontal forces at the abutments, which determines the economy of that solution in many cases.
required to increase stiffness under traffic loads, but to guarantee serviceability, i.e. respect the maximum longitudinal slope.
(wheelchairs, bike routes), a sag/span ratio of f/l < 1/67 is required, i.e., more than six times less than in a typical suspension bridge. Even for narrow footbridges, this results in very high cable forces.
z x
k
q
f l
2
Cable force: 8 , 4 2 4 Maximum slope: Example: (2.5 m wide bridge, dead load only considered to verify maximum slope ): 100 m, 6%, 20 kN/m 1.50 m
k k adm adm adm adm k
q l H q l l f f z H z f z l z l z q f = → = = = → → 16.7 MN H
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the sag is very small, their shape is almost exactly a second
for the combined cable-type and bending response. As outlined by Marti (Theory of Structures, 2013, Chapter 18.9), the differential equation with the solution covers the entire spectrum from a pure bending response only (l = 0), to a pure cable-type response (l → ). Note that it has been assumed in the derivation of the differential equation that the dead load g is carried by cable tension alone.
suspension bridges, capable of accounting for large deformations, are used for detailed design.
( ) ( )
( ), ( ) H EIw H H w q g H H g H H q H − + = − = =
1 2 3 4
cosh( ) sinh( )
part
H H w c c x c x c x w EI + = + + l + l + l =
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neglecting the bending stiffness (as cables). The cable equation (see section static analysis of cables) can be applied directly.
and service stages, see figure.
erection, after hardening of the concrete joints (i.e., change from cable to stress-ribbon behaviour) – is decisive for the stresses in the structure during all future stages.
dead load configuration after prestressing (step e). Since this geometry depends on the basic stage, iterative calculations are required.
due to traffic loads and thermal effects (reduced sag at cold temperature causes higher cable forces under same vertical load).
prestressing
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constant in all spans to avoid large horizontal loads on the intermediate piers. A constant horizontal force (under equal load) corresponds to sags proportional to the squared length
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bending moments near the supports (abutments, piers), where large curvatures occur, mainly due to prestressing and thermal effects.
→ supporting the stress-ribbon on a saddle from which it can lift during post-tensioning and temperature drop, and to which the band can return for a temperature increase (left figures) → Strengthening the stress-ribbon with a short support haunch (right figures), which will, however in turn attract higher moments.
End support details for stress-ribbon bridges [Strasky 2004] Pier support details for stress-ribbon bridges [Strasky 2004]
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hinges (preferably concrete hinges for low maintenance) may be provided at the foot (figure).
Pier of Prague-Troja stress-ribbon bridge [Strasky 2004]
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are integral parts of the abutments.
to transfer very high horizontal forces to the soil. Anchorage is commonly done by → Rock or ground anchors (combined with micropiles acting in compression at the front of the abutment except in very stiff soil at foundation level) → Micropiles forming a “triangulation” (lower figure)
anchors may therefore have to be stressed in two stages (e.g. 50% initially, full prestress after activation of stress-ribbon self weight).
End anchorage by rock anchors [Strasky 2004] End anchorage by micropiles [Strasky 2004]
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shoring, since installation is done using bearing tendons (which are activated for ULS design, no temporary cables).
→ place segment under bearing tendons → hang segment to bearing tendons (such that it can slide)
a winch and attach to previously installed segments
at saddles
cannot be directly installed along them, requiring temporary cables and a trolley for segment installation.
Typical cross-section [Strasky 2004]